Financial models with long-tailed distributions and volatility clustering have been introduced to overcome problems with the realism of classical financial models.
These classical models of financial time series typically assume homoskedasticity and normality and as such cannot explain stylized phenomena such as skewness, heavy tails, and volatility clustering of the empirical asset returns in finance.
, the tempered stable processes have been proposed for overcoming this limitation of the stable distribution.
On the other hand, GARCH models have been developed to explain the volatility clustering.
In the GARCH model, the innovation (or residual) distributions are assumed to be a standard normal distribution, despite the fact that this assumption is often rejected empirically.
For this reason, GARCH models with non-normal innovation distribution have been developed.
Many financial models with stable and tempered stable distributions together with volatility clustering have been developed and applied to risk management, option pricing, and portfolio selection.
is called infinitely divisible if, for each
, there are independent and identically-distributed random variables such that where
denotes equality in distribution.
is called a Lévy measure if
is infinitely divisible, then the characteristic function
is called a Lévy triplet of
satisfying the conditions above, there exists an infinitely divisible random variable
A real-valued random variable
All stable random variables are infinitely divisible.
A stable random variable
An infinitely divisible distribution is called a classical tempered stable (CTS) distribution with parameter
This distribution was first introduced by under the name of Truncated Lévy Flights[1] and 'exponentially truncated stable distribution' [2].
It was subsequently called the tempered stable or the KoBoL distribution.
for a tempered stable distribution is given by for some
The KR distribution, which is a subclass of the Rosiński's generalized tempered stable distributions, is used in finance.
[5] An infinitely divisible distribution is called a modified tempered stable (MTS) distribution with parameter
is the modified Bessel function of the second kind.
The MTS distribution is not included in the class of Rosiński's generalized tempered stable distributions.
[6] In order to describe the volatility clustering effect of the return process of an asset, the GARCH model can be used.
In the GARCH model, innovation (
For that reason, new GARCH models with stable or tempered stable distributed innovation have been developed.
[7][8][9] Subsequently, GARCH Models with tempered stable innovations have been developed.
[6][10] Objections against the use of stable distributions in Financial models are given in [11][12]